Abstract
For convenience, all spaces considered here will be over the complex field. Given any Banach spaces X and Y, L (X, Y) will stand for the space of all bounded linear maps from X to Y; we shall write L(X) instead of L(X, X). We also put B X = {x ∈ X : ‖x‖ ≤ 1}. Let T ∈ L(X, Y) and n ∈ ℕ.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Amick, C.J. Some remarks on Rellich’s theorem and the Poincaré inequality. J. London Math. Soc. 18 (1978), 81–93.
Birman, M.S. and Solomjak, M.Z. Spectral asymptotics of non-smooth elliptic operators, I. Trans. Moscow Math. Soc., 27 (1972), 1–52.
Birman, M.S. and Solomjak, M.Z. Spectral asymptotics of non-smooth elliptic operators, II. Trans. Moscow Math. Soc., 28 (1973), 1–32.
Birman, M.S. and Solomjak, M.Z. Quantitative analysis in Sobolev imbedding theorems and applications to spectral theory. Amer. Math. Soc. Transl., 114 (1980), 1–132.
Bourgain, J., Pajor, A., Szarek, S.J., and Tomczak-Jaegermann, N. On the duality problem for entropy numbers of operators. In Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, Vol. 1376. Springer-Verlag, New York, 1989.
Carl, B. Entropy numbers, s-numbers and eigenvalue problems. J. Funct. Anal., 41 (1981), 290–306.
Carl, B. and Stephani, I. Entropy, Compactness and the Approximation of Operators. Cambridge University Press, Cambridge, 1990.
Carl, B. and Triebel, H. Inequalities between eigenvalues, entropy numbers and related quantities in Banach spaces. Math. Ann., 251 (1980), 129–133.
Edmunds, D.E., Edmunds, R.M., and Triebel, H. Entropy numbers of embeddings of fractional Besov—Sobolev spaces in Orlicz spaces. J. London Math. Soc., 35 (1987), 121–134.
Edmunds, D.E. and Evans, W.D. Spectral Theory and Differential Operators. Oxford University Press, Oxford, 1987.
Edmunds, D.E. and Opic, B. Weighted Poincaré and Friedrichs inequalities. J. London Math. Soc. 47 (1993), 79–93.
Edmunds, D.E., Opic, B., and Pick, L. Poincaré and Friedrichs inequalities in abstract Sobolev spaces. Math. Proc. Cambridge Philos. Soc., 113 (1993), 355–379.
Edmunds, D.E., Opic, B., and Rákosník, J. Poincaré and Friedrichs inequalities in abstract Sobolev spaces, II. Math. Proc. Cambridge Philos. Soc., 115 (1994), 159–173.
Edmunds, D.E. and Triebel, H. Entropy numbers and approximation numbers in function spaces. Proc. London Math. Soc., 58 (1989), 137–152.
Edmunds, D.E. and Triebel, H. Entropy numbers and approximation numbers in function spaces, II. Proc. London Math. Soc., 64 (1992), 153–169.
Edmunds, D.E. and Triebel, H. Function Spaces, Entropy Numbers, Differential Operators. Cambridge University Press, Cambridge, 1996.
Edmunds, D.E. and Tylli, H.-O. On the entropy numbers of an operator and its adjoint. Math. Nachr., 126 (1986), 231–239.
Edmunds, D.E. and Tylli, H.-O. Entropy numbers of tensor products of operators. Ark. Mat., 31 (1993), 247–274.
Evans, W.D. and Harris, D.J. Sobolev embeddings for ridged domains. Proc. London Math. Soc., 54 (1987), 141–176.
Haroske, D., and Triebel, H. Entropy numbers in weighted function spaces and eigenvalue distributions of some degenerate pseudodifferential operators, I. Math. Nachr., 167 (1994), 131–156; II, ibid, 168 (1994), 109–137.
König, H. On the tensor stability of s-number ideals. Math. Ann., 269 (1984), 77–93.
König, H. Eigenvalue Distribution of Compact Operators. Birkhäuser, Basel, 1986).
Pietsch, A. Operator Ideals. North-Holland, Amsterdam, 1980.
Pietsch, A. Tensor products of sequences, functions and operators. Arch. Math., 38 (1982), 335–344.
Strichartz, R.S. A note on Trudinger’s extension of Sobolev’s inequality. Indiana Univ. Math. J., 21 (1972), 841–842.
Triebel, H. Theory of Function Spaces. Birkhäuser, Basel, 1983.
Triebel, H. Theory of Function Spaces, II. Birkhäuser, Basel, 1992.
Triebel, H. Approximation numbers and entropy numbers of embeddings of fractional Besov-Sobolev spaces in Orlicz spaces. Proc. London Math. Soc., 66 (1993), 589–618.
Triebel, H. Relations between approximation numbers and entropy numbers. J. Approx. Theory, 78 (1994), 112–116.
Trudinger, N.S. On imbedding into Orlicz spaces and some applications. J. Math. Mech., 17 (1967), 473–484.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1998 Springer-Verlag New York, Inc.
About this chapter
Cite this chapter
Edmunds, D.E. (1998). Entropy Numbers, Approximation Numbers, and Embeddings. In: Buttazzo, G., Galdi, G.P., Lanconelli, E., Pucci, P. (eds) Nonlinear Analysis and Continuum Mechanics. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-2196-8_4
Download citation
DOI: https://doi.org/10.1007/978-1-4612-2196-8_4
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4612-7455-1
Online ISBN: 978-1-4612-2196-8
eBook Packages: Springer Book Archive